$E$ is an elliptic curve with non-split multiplicative reduction at prime $p$. I'm trying to find the number of points $E$ over $F_{p^n}$.
I know that when I remove the singularity, the rest is a group isomorphic to the kernel of the norm map from the quadratic extension of the base, t.i. includes $p^n+1$ elements (if I'm not mistaken). Plus one singular point and the answer is $p^n+2$.
But I found that this number of points depends whether n is odd or even. Why it is so?